Intro Chapter

So I am pretty excited.  I had originally bought the book How to Prove It though Google online books. Unfortunately it was missing the intro chapter and there were a bunch other issues that made using it a pain (thought the fact that I have it on my iPad is cool since I pretty much have my iPad with me at all times).

Anyway I just finished reading the Intro chapter and have started working on some of the problems and I have to say, this book is so freaking cool.  Just the exposure to some really cool math stuff that I have never seen before is totally worth the cover cost.

Couple highlights.  So there are prime numbers of the form $$ 2^n-1 $$ where n is a prime.  They are called Mersenne primes after a guy called, well of course by a monk (I think he was a monk now I am not so certain) named Mersenne (1588-1647).  Mersenne primes are currently some of the largest prime number numbers in fact, the largest know prime is a Mersenne prime.

Anyway, that's kind of cool in a did you know this minutia kind of way.  Here is the cool thing our buddy Euclid proved that if a prime number is of the form $$ 2^n-1 $$ then $$ 2^{n-1}(2^n-1) $$ is perfect.  Meaning every Mersenne prime can be used to generate a perfect number.

Note a perfect number is a number n such that the sum of all its factors less than n = n.  So for example 28 is a perfect number because its factors: 1,2,4,7,14 summed up equal 28.

What really got me though was later on Euler (1707-1783) proved that every perfect number is generated this way.  So again using 28 as an example.  $$28 = 2^2(2^3-1) $$.

So those were all pretty cool things.  Here is something cool you can do with non prime numbers of the form $$ 2^n-1 $$ .  If n is not prime then as a consequence $$ 2^n-1 $$ is also not prime.  As a result we can take non Mersenne prime numbers of that form and factor them (if of course you know how to write them as $$ 2^n-1 $$ which I will agree is no mean feet and something I would really like to know how to do - maybe this would be a nice little programming project in the very near future.

Anyway, lets take a perfect number of this form, for example the one given in the book: 32,767 which equals $$ 2^{15}-1 $$, then we can two of its factors.

$$ 2^{15}-1 = 2^{3 \cdot 5}-1 $$
$$ = (2^3-1)(1+2^{3}+2^{3\cdot2}+...+2^{3\cdot4} $$

The general formula for this is:  if $$n=a\cdot b $$ then $$2^n-1= 2^{a \cdot b}-1$$
$$ = (2^a-1)(1+2^a+2^{a \cdot 2}+...+2^{a \cdot (b-1)}) $$

Pretty cool right.

Chapter 1: Deductive Reasoning and Logical Connectives

Question:
1.  Analyze the logical forms of the following statements:
(a) We'll have either a reading assignment or homework problems, but we won't have both homework problems and a test.
(b) You won't go skiing, or you will and there won't be any snow.
(c) $\sqrt{7} \nleq2 $

Answer:
R = We have a reading assignment.
Q = We have a homework assignment
R = We have a test

 2. Analyze the logical forms of the following statements:
(a) Either John and Bill are both telling the truth, or neither of them is.
(b) I'll have either fish or chicken, but I won't have both fish and mashed potatoes.
(c) 3 is a common divisor or 6,9, and 15.

 3. Analyze the logical forms of the following statements:
(a) Alice and Bob are not both in the room.
(b) Alice and Bob are both not in the room.
(c) Either Alice or Bob is not in the room.
(d) Neither Alice nor Bob is in the room.

 5. Let P stand for the statement "I will buy the pants" and S for the statement "I will buy the shirt.". What English sentences are represented by the following expressions.

(a) $\neg (P \wedge \neg S)$
(b) $\neg P \wedge \neg S$
(c) $\neg P \vee \neg S$

Purpose of This Blog

I am reading through the book: How to Prove It by Daniel J. Velleman and blogging my solutions and thoughts about the exercises given and the concepts presented.